Spectral Shadows: When Communication Complexity Meets Linear Invariance Testing

TL;DR

This work studies the LinIso_n problem—testing linear isomorphism of Boolean functions under invertible domain transformations—in a communication complexity setting. It reveals that the approximate spectral norm $t$ of the inputs fundamentally governs complexity, and develops both deterministic and private-coin randomized protocols with costs polynomial in $t$ (deterministic: $O(t^4\log^2(1/\omega))$, private-coin: $O(t^2/\omega^3)$) alongside nearly matching lower bounds. A key technical contribution is new junta-type results for functions with small approximate spectral norm, enabling structural compression and efficient testing. The results establish approximate spectral norm as a central complexity measure for linear invariance testing in communication models and open avenues for refining bounds and exploring tolerant versions. The work integrates Fourier-analytic techniques with complexity-theoretic reductions to advance our understanding of symmetry-testing problems in communication settings.

Abstract

In this short note, we initiate the study of the Linear Isomorphism Testing Problem in the setting of communication complexity, a natural linear algebraic generalization of the classical Equality problem. Given Boolean functions $f, g : \mathbb{F}_2^n \to \{-1, +1\}$, Alice and Bob are tasked with determining whether $f$ and $g$ are equivalent up to a nonsingular linear transformation of the input variables, or far from being so. This problem has been extensively investigated in several models of computation, including standard algorithmic and property testing frameworks, owing to its fundamental connections with combinatorial circuit design, complexity theory, and cryptography. However, despite its broad relevance, it has remained unexplored in the context of communication complexity, a gap we address in this work. Our main results demonstrate that the approximate spectral norm of the input functions plays a central role in governing the communication complexity of this problem. We design a simple deterministic protocol whose communication cost is polynomial in the approximate spectral norm, and complement it with nearly matching lower bounds (up to a quadratic gap). In the randomised setting with private coins, we present an even more efficient protocol, though equally simple, that achieves a quadratically improved dependence on the approximate spectral norm compared to the deterministic case, and we prove that such a dependence is essentially unavoidable. These results identify the approximate spectral norm as a key complexity measure for testing linear invariance in the communication complexity framework. As a core technical ingredient, we establish new junta theorems for Boolean functions with small approximate spectral norm, which may be of independent interest in Fourier analysis and learning theory.

Theorems & Definitions (16)

• Definition 1.1: The $\mathsf{LinIso}_n$ problem
• Theorem 1.2: Deterministic Communication Complexity
• Theorem 1.3: Randomized Communication Complexity
• Lemma 2.1: Approximate Spectral Sampling
• Theorem 2.2: Hoeffding's inequality
• proof : Proof of Lemma \ref{['lemma:BS approximation']}
• proof : Proof of Theorem \ref{['theorem:deterministic-upper-lower']} (a)
• Remark 2.3
• Definition 2.4: $\mathsf{EQU}_n$
• Definition 2.5: ${\widetilde{\mathsf{LinIso}}}_{\ell}$